A Feynman-Kac Formula for the Subcritical Ultraviolet-Renormalized Spin Boson Model

TL;DR

This paper establishes a Feynman-Kac formula for a renormalized spin boson model, enabling probabilistic analysis of the system and demonstrating ground state persistence without an ultraviolet cutoff.

Contribution

It introduces a Feynman-Kac formula for the self-energy renormalized spin boson Hamiltonian, extending probabilistic methods to this quantum field model.

Findings

Proves a Feynman-Kac formula for the renormalized spin boson Hamiltonian.

Shows ground state existence persists after removing ultraviolet cutoff.

Provides a probabilistic framework for analyzing the spin boson system.

Abstract

We prove a Feynman-Kac formula (FKF) for the self-energy renormalized spin boson Hamiltonian, describing a two-state quantum system linearly coupled to a bosonic quantum field. Similar to recent FKFs for the Fr\"ohlich polaron and the non- and semi-relativistic Nelson models, it yields a probabilistic treatment of the spin as a jump process, but treats the field on the usual bosonic Fock space. As an application, we prove that the existence of ground states for infrared-regular models persists the removal of an ultraviolet cutoff.